Families escaping the fighting in the Iraqi city of Fallujah said Islamic State was forcing them out of their homes and moving them from house to house in the crossfire.

"We tried going back to our house, but when we saw lots of other families fleeing, we joined them," said one woman who gave her name as Suad.

"I was carrying my 2-year-old daughter, Hana, as I ran barefoot to reach the other families," Suad told the Norwegian Refugee Council.

According to the NRC, which is providing aid to the refugees, Suad, her husband Ali and their six children, finally made it to the al-Iraq camp some 30 kilometers from Fallujah on May 25.

Iraqi forces, backed by coalition airstrikes and Shi'ite militias, have ringed most of Fallujah. Counterterrorism units were said to have entered the city on Monday.

In pictures: Families Fleeing Fallujah War Zone

Families Flee Fallujah War Zone

A U.S. State Department official tells VOA the Untied States is "supporting the ISF's efforts to protect civilians, including efforts to create safe passageways for civilians in Fallujah to flee."

Islamic State has controlled, a Sunni city, for more than two years.

Iraqi families are seen near al-Sejar village in Anbar province, after fleeing the city of Fallujah, May 27, 2016, during a major operation by pro-government forces to retake the city from the Islamic State (IS) group.

One big prison

Jabar Muhammad Odaa, who ran away from the city six months ago when it was under siege, said that under Islamic State rule, Fallujah had become one big prison.

"They would not let anyone leave," he told VOA in a phone interview from the al-Iraq camp. "We lived like animals."

"They [IS] took everything from us; it was like a big prison. They said, 'if you want to leave, you leave, but your family stays here,'" said Odaa. "They put the children and young boys in the mosque, and said if you join us you and your family will be better off. Every single Friday they pushed me to go to the mosque and learn with them."

Odaa said men who shaved their beards or mustaches would get 100 lashes. His descriptions echoed those of other families who escaped Islamic State earlier this year in the northern Mosul area.

Although many foreign fighters are believed to be with IS, Odaa said he only saw Iraqis among the extremists.

IS appears to be using fighting tactics that have been found before in Ramadi and villages around Mosul, such as building tunnels and lacing the area with bombs.

"They have built tunnels under the road between Amariyat and Fallujah," he said, referring to a route that links Fallujah to a city further south, where refugee camps are being set up.

Getting out

Becky Bakr Abdulla, media coordinator for NRC in Iraq, said more than 530 families have managed to escape the fighting.

Civilians who fled their homes due to the clashes on the outskirts of Falluja, gather in the town of Garma, Iraq, May 30, 2016.

Only one family since Monday, however, has managed to leave from the center of the city, where the fighting is now the most intense, she said.

“Families we have been in touch with point to especially two factors that are preventing them from leaving,” Abdulla told VOA in a telephone interview. “They say it is too dangerous for them to leave their houses, and that, ‘IS came to our doors and threatened us.’”

Smoke billows through the air during an offensive by Iraqi military forces into Fallujah to retake the city from Islamic State militants in Iraq, in an image taken from video, May 30, 2016.

Humanitarian agencies have repeatedly asked for all warring sides to allow civilians safe passage away from the fighting. Some 50,000 civilians are believed to still be trapped in the city.

“Everyone I am talking to is telling me that the only things they survived on are dates, and drinking water from the river. There is no medicine, no fuel, no electricity; people are staying indoors. The situation is extremely critical,” Abdulla said.

WATCH: UNHCR concerned about Fallujah residents

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Sharon Behn is a foreign correspondent working out of Voice of America’s headquarters in Washington D.C Her current beat focuses on political, security and humanitarian developments in Iraq, Syria and Turkey. Follow Sharon on Twitter and on Facebook.